Predictable process

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In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes.

Mathematical definition

Discrete-time process

Given a filtered probability space ( Ω , F , ( F n ) n ∈ N , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },\mathbb {P} )} {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },\mathbb {P} )}, then a stochastic process ( X n ) n ∈ N {\displaystyle (X_{n})_{n\in \mathbb {N} }} {\displaystyle (X_{n})_{n\in \mathbb {N} }} is predictable if X n + 1 {\displaystyle X_{n+1}} {\displaystyle X_{n+1}} is measurable with respect to the σ-algebra F n {\displaystyle {\mathcal {F}}_{n}} {\displaystyle {\mathcal {F}}_{n}} for each n.[1]

Continuous-time process

Given a filtered probability space ( Ω , F , ( F t ) t ≥ 0 , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )} {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}, then a continuous-time stochastic process ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} {\displaystyle (X_{t})_{t\geq 0}} is predictable if X {\displaystyle X} {\displaystyle X}, considered as a mapping from Ω × R + {\displaystyle \Omega \times \mathbb {R} _{+}} {\displaystyle \Omega \times \mathbb {R} _{+}}, is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.[2] This σ-algebra is also called the predictable σ-algebra.

Examples

See also

References

  1. van Zanten, Harry (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (PDF). Archived from the original (pdf) on April 6, 2012. Retrieved October 14, 2011.
  2. "Predictable processes: properties" (PDF). Archived from the original (pdf) on March 31, 2012. Retrieved October 15, 2011.